3.1 New compounding construction
3.1.1 Compounding Kn¨odel graphs with binomial trees
In this section, we introduce a new broadcast graph construction similar to the com- pounding method in [3] for any 2m−1 + 1 ≤ n ≤ 2m− 1, where m ≥ 5, but using Kn¨odel
graphs as a base instead of a hypercube. The later comparison shows that this construction improve the upper bound on B(n) for any 2m−1 + 1 ≤ n ≤ 2m − 2f rac12(m+3), where
n = 2m− 2k− d, m ≥ 5, 2 ≤ k ≤ m − 2, and 0 ≤ d ≤ 2k− 1. It is clear that any value of
n ∈ [2m−1+ 1, 2m− 2] can be represented as n = 2m− 2k− d, where 1 ≤ k ≤ m − 2 and
0 ≤ d ≤ 2k− 1. For convenience, we let l = k − 1, n = 2m − 2l+1− d, 0 ≤ l ≤ m − 3,
and 0 ≤ d ≤ 2l+1− 1 in the following constructions.
The new broadcast graph L = (V, E) on n = (2m−l − 2)2l vertices, where m ≥ 5 and
by BT0,BT1,..., BT2m−l−3. The roots of the binomial trees denoted by ri, form a Kn¨odel
graph KG2m−l−2 on 2m−l − 2 vertices, 0 ≤ i ≤ 2m−l − 3. Figure3.1 presents the new
construction for m = 6 and l = 2.
The next step of the construction is to delete d vertices from L, where 0 ≤ d ≤ 2l+1− 1, in
order to obtain any 2m−1+ 1 ≤ n ≤ 2m− 1, the given number of vertices of the broadcast
graph, where m ≥ 5. This step can be done by deleting a leaf from any binomial tree repeatedly. Note that we do not delete the root of any binomial tree because it also belongs to KG2m−l−2. The number of deleted vertices is at most 2l+1− 1.
Then the new construction connects the vertices of binomial trees BT0, BT1,..., BT2m−l−3
to m − l − 1 vertices of KG2m−l−2.
Let ri be the root of binomial tree Bi and rh be the first dimensional neighbor of ri in
KG2m−l−2. By the definition of Kn¨odel graph, h ≡ 1 − i mod 2m−l− 2. We connect each
non-root vertex w in binomial tree BTi to all the neighbors of rh in KG2m−l−2. Let rj de-
note these neighbors, j +h ≡ j +1−i ≡ 2s−1 mod 2m−l−2 for all s = 1, 2, ..., m−l−1.
The edges of E of graph L are of three types: the edges in the Kn¨odel graph KG2m−l−2
denoted by EH, the edges in all binomial trees BT0, BT1, ..., BT2m−l−3denoted by ET, and
the edges between vertex w ∈ BTi and some vertices in the Kn¨odel graph denoted by EP.
Therefore, the set of edges of graph L = (V, E) is defined as E = EH ∪ ET ∪ EP, where
EP = {(w, rj)|j + 1 − i ≡ 2s− 1 mod 2m−l− 2, 1 ≤ s ≤ m − l − 1, w ∈ BTi\ {ri}, rj ∈
KG2m−l−2}. Thus, the number of edges in L is |E| = |EH| + |ET| + |EP|. The Kn¨odel
graph KG2m−l−2has
|EH| =
(m − l − 1)(2m−l− 2)
2
edges. All 2m−l− 2 binomial trees BT0, BT1, ...BT2m−l−3together have
tree edges. To count the number of edges in EP, each binomial tree has 2l− l − 1 vertices
except the root and its l neighbors on the first level. In total, graph L has (2m−l− 2)(2l−
l − 1) − d such vertices remaining after removing d leaves. Each of these vertices needs m − l − 1 edges to connect to the vertices in the Kn¨odel graph. And each of the vertices on the first level of any binomial tree (the l neighbors of the root within a binomial tree) needs m − l − 2 additional edges connecting to the vertices of KG2m−l−2, since it is already
adjacent to its root. Thus,
|EP| = ((2m−l− 2)(2l− l − 1) − d)(m − l − 1) + (2m−l− 2)l(m − l − 2)
The total number of edges of graph L is
|E| = (m − l)n − (m + l + 1)2m−l−1+ m + l + 1
In summary, graph L has |V | = n vertices for any n = 2m− 2l+1− d, where 0 ≤ l ≤ m − 3
and 0 ≤ d ≤ 2l+1− 1, 2m−l− 2 vertices and edges of KG
2m−l−2, and every vertex of any
binomial tree BTi, 0 ≤ i ≤ 2m−l− 2 is connected to m − l − 1 vertices of KG2m−l−2.
Figure3.1 demonstrates our construction of graph L for l = 2, m = 6, and 0 ≤ d ≤ 7. We first construct a Kn¨odel graph on 24− 2 vertices. The vertices of KG
14are labeled as
r0, r1, r2, ..., r13. Each vertex of KG14is attached a binomial tree on 4 vertices. Then, for
example, we connect vertex w ∈ BT0to root vertices r0, r2and r6, which are the neighbors
B0 B1 B2 B3 B4 B5 B6 B7 B8 B9 B10 B11 B12 r0 r1 r2 r3 r4 r5 r6 r7 r8 r9 r10 r11 r12 r13 w B13
Figure 3.1: An example of L, when m − l = 4. Solid lines and vertices ri form the
Kn¨odel graph KG14. Each binomial tree of degree 2 is replaced by a dotted triangle. A tree
vertex w of binomial tree BT0 and the dashed edges show an example of the connections
between a non-root vertex and the root vertices. w is connected to the neighbors of the first dimensional neighbor of the root vertex of tree BT0.
Theorem 3.1. L is a broadcast graph and for any n = 2m − 2l+1 − d, where m ≥ 5,
1 ≤ l ≤ m − 3, and 0 ≤ d ≤ 2l+1− 1
B(n) ≤ (m − l)n − (m + l + 1)2m−l−1+ m + l + 1
Proof. It is clear that n ∈ [2m−1+ 1, 2m− 2] for any n above. Thus, dlog ne = m. To show that L is a broadcast graph, broadcast scheme for any originator is described below.
(1) If the originator is a root vertex ri in KG2m−l−2, where 0 ≤ i ≤ 2m−l − 3, then
the broadcast scheme of ri consists of the broadcast scheme from originator ri in
KG2m−l−2 concatenated with the broadcast scheme in all binomial tree from their
roots. ri first completes broadcasting within the Kn¨odel graph using dimensional
broadcast scheme by time unit m − l. So, after time m − l the roots of all binomial trees have the message. Then it takes l time units to broadcast in its binomial tree. Thus, the broadcasting in L completes in m time units.
(2) If the originator is a non-root vertex w in BTi, 0 ≤ i ≤ 2m−l− 3; the broadcasting
is more complicated. By our construction, w is adjacent to all the neighbors of rh,
which is the first dimensional neighbor of ri - the root of binomial tree BTi.
Consider the dimensional broadcast scheme of Kn¨odel graphs from rhin KG2m−l−2.
rhinforms its neighbor on dimension t at time unit t for all t = 1, 2, ..., m − l. Since
w is adjacent to all neighbors of rh, w can play the role of rhin the broadcast scheme
from originator w in L. w informs the i-th dimensional neighbor of vertex rh at time
unit i, for all i = 1, 2, ..., m − l − 1. Every informed vertex continues broadcasting as in the dimensional broadcast scheme from the originator rh. As a result, every vertex
in KG2m−l−2 except rh can be informed by the same broadcast scheme from rh in
KG2m−l−2at the same time, which is m − l. Then rh can be informed by a call from
riat time unit m − l. Note that since the degree of vertex riin KG2m−l−2is m − l − 1
and ri is busy during the first m − l − 1 time units, then ri is idle at time unit m − l,
and so it can call vertex rh. The first m − l time units of the broadcast scheme from
... ... w u1 (ri) u2 u3 ut um-l-1 rh KG2 m-l-2 m-l-1 m-l-1
1
2
3
t
Figure 3.2: The broadcast scheme from w in L in the first m − l time units. ut, 1 ≤ t ≤
m − l − 1 is t dimensional neighbor of rh. Solid arcs denote the calls of the broadcast
scheme from originator rh in KG2m−l−2. Dashed arcs denote the calls from originator w
in L. All the other calls of the broadcast scheme from originator rh in KG2m−l−2, and the
broadcast scheme of originator w in graph L are the same. The numbers besides the arcs are the times of calls.
Now, every vertex rj, 1 ≤ j ≤ 2m−l− 3 in KG2m−l−2, which is also the root of BTj,
is informed after time m − l. Next, every root rj broadcasts all vertices within its
respective binomial tree in the remaining l time units. The broadcasting in L again takes m time units in total.
Therefore, L is a broadcast graph. And for any n = 2m − 2l+1− d ∈ [2m−1+ 1, 2m − 2],
where m ≥ 5, 0 ≤ l ≤ m − 3, and 0 ≤ d ≤ 2l+1− 1
B(n) ≤ (m − l)n − (m + l + 1)2m−l−1+ m + l + 1
Theorem 3.2.
B(n) ≤ (m − k + 1)n − (m + k)2m−k+ m + k,
where n = 2m− 2k− d, m ≥ 5, 1 ≤ k ≤ m − 2, and 0 ≤ d ≤ 2k− 1